How to animate object target positions while respecting rigid body collisions?
The solution, indicated by Rich Sedman (thank you!), is to use Rigid Body Constraints with Generic Springs.


Create 3 cubes
Add Rigid Body to each cube
Create 3 spheres (will act as targets)
Add Rigid Body (passive) to each sphere. Mark it as animated and place it in Collections 2 to avoid colliding with the boxes.
Add Rigid Body Constraint to each sphere, type Generic Spring. Activate Linear limits in all 3 axes. For Objects set First to the sphere itself, Second to the corresponding cube. Set Stiffness and Damping for Springs &gt; Angular and Linear according to taste.
Put each sphere centered in a cube. If you place the cube under the sphere, it will be hanging. When you animate the sphere position the cube will follow, hanging under it. I wanted to avoid that, so it doesn't feel like marionettes.
Animate the sphere positions so they pass near other spheres. The Cubes will collide and avoid each other.
Disable gravity (optional).


What's missing for me is to figure out the right values for the constraints so the motion feels right.